In fact, \(\mathcal {H}^{p}_{FIO}(\mathbb {R}^{d})\)  was originally defined in [1]}, [2]} using conical square function estimates over the cosphere bundle. This includes an intrinsic definition of \(\mathcal {H}^{\infty }_{FIO}(\mathbb {R}^{d})\)  in terms of Carleson measures. The equivalent characterization in Definition REF  was obtained in [3]}, [4]}.
